Spherically symmetric equilibria for self-gravitating kinetic or fluid models in the non-relativistic and relativistic case - A simple proof for finite extension

TL;DR

This paper provides a unified, simple proof demonstrating that spherically symmetric equilibria in various self-gravitating models have finite extension, based on local growth conditions of the equation of state.

Contribution

It introduces a unified, straightforward proof for finite extension of equilibria across multiple self-gravitating models, generalizing known local growth conditions.

Findings

Proves finite extension for all considered models.

Characterizes equilibria via local growth conditions.

Applicable to both non-relativistic and relativistic cases.

Abstract

We consider a self-gravitating collisionless gas as described by the Vlasov-Poisson or Einstein-Vlasov system or a self-gravitating fluid ball as described by the Euler-Poisson or Einstein-Euler system. We give a simple proof for the finite extension of spherically symmetric equilibria, which covers all these models simultaneously. In the Vlasov case the equilibria are characterized by a local growth condition on the microscopic equation of state, i.e., on the dependence of the particle distribution on the particle energy, at the cut-off energy E_0, and in the Euler case by the corresponding growth condition on the equation of state p=P(\rho) at \rho=0. These purely local conditions are slight generalizations to known such conditions.